Supply Our Soldiers Act of 2019 This bill provides postal benefits to certain active duty service members. It directs the Department of Defense to provide for a program under which vouchers (used for postal-free mailings) are provided to members of the Armed Forces who are on active duty and who are either (1) serving in Iraq, Afghanistan, or another hostile fire area designated as eligible for hazardous duty pay; or (2) hospitalized at a military medical facility as a result of such service. The benefits program shall apply during FY2020-FY2025.